3.70.29 \(\int \frac {-24+546 x^2+3380 x^4+(-1+47 x^2+260 x^4) \log (\frac {1+5 x^2}{x^2})+(x^2+5 x^4) \log ^2(\frac {1+5 x^2}{x^2})}{e^4 (676 x^2+3380 x^4)+e^4 (52 x^2+260 x^4) \log (\frac {1+5 x^2}{x^2})+e^4 (x^2+5 x^4) \log ^2(\frac {1+5 x^2}{x^2})} \, dx\)

Optimal. Leaf size=20 \[ \frac {x+\frac {1}{x+x \left (25+\log \left (5+\frac {1}{x^2}\right )\right )}}{e^4} \]

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Rubi [F]  time = 0.83, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-24+546 x^2+3380 x^4+\left (-1+47 x^2+260 x^4\right ) \log \left (\frac {1+5 x^2}{x^2}\right )+\left (x^2+5 x^4\right ) \log ^2\left (\frac {1+5 x^2}{x^2}\right )}{e^4 \left (676 x^2+3380 x^4\right )+e^4 \left (52 x^2+260 x^4\right ) \log \left (\frac {1+5 x^2}{x^2}\right )+e^4 \left (x^2+5 x^4\right ) \log ^2\left (\frac {1+5 x^2}{x^2}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-24 + 546*x^2 + 3380*x^4 + (-1 + 47*x^2 + 260*x^4)*Log[(1 + 5*x^2)/x^2] + (x^2 + 5*x^4)*Log[(1 + 5*x^2)/x
^2]^2)/(E^4*(676*x^2 + 3380*x^4) + E^4*(52*x^2 + 260*x^4)*Log[(1 + 5*x^2)/x^2] + E^4*(x^2 + 5*x^4)*Log[(1 + 5*
x^2)/x^2]^2),x]

[Out]

x/E^4 + (2*Defer[Int][1/(x^2*(1 + 5*x^2)*(26 + Log[5 + x^(-2)])^2), x])/E^4 - Defer[Int][1/(x^2*(26 + Log[5 +
x^(-2)])), x]/E^4

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-24+546 x^2+3380 x^4+\left (-1+47 x^2+260 x^4\right ) \log \left (5+\frac {1}{x^2}\right )+\left (x^2+5 x^4\right ) \log ^2\left (5+\frac {1}{x^2}\right )}{e^4 x^2 \left (1+5 x^2\right ) \left (26+\log \left (5+\frac {1}{x^2}\right )\right )^2} \, dx\\ &=\frac {\int \frac {-24+546 x^2+3380 x^4+\left (-1+47 x^2+260 x^4\right ) \log \left (5+\frac {1}{x^2}\right )+\left (x^2+5 x^4\right ) \log ^2\left (5+\frac {1}{x^2}\right )}{x^2 \left (1+5 x^2\right ) \left (26+\log \left (5+\frac {1}{x^2}\right )\right )^2} \, dx}{e^4}\\ &=\frac {\int \left (1+\frac {2}{x^2 \left (1+5 x^2\right ) \left (26+\log \left (5+\frac {1}{x^2}\right )\right )^2}-\frac {1}{x^2 \left (26+\log \left (5+\frac {1}{x^2}\right )\right )}\right ) \, dx}{e^4}\\ &=\frac {x}{e^4}-\frac {\int \frac {1}{x^2 \left (26+\log \left (5+\frac {1}{x^2}\right )\right )} \, dx}{e^4}+\frac {2 \int \frac {1}{x^2 \left (1+5 x^2\right ) \left (26+\log \left (5+\frac {1}{x^2}\right )\right )^2} \, dx}{e^4}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.19, size = 20, normalized size = 1.00 \begin {gather*} \frac {x+\frac {1}{x \left (26+\log \left (5+\frac {1}{x^2}\right )\right )}}{e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-24 + 546*x^2 + 3380*x^4 + (-1 + 47*x^2 + 260*x^4)*Log[(1 + 5*x^2)/x^2] + (x^2 + 5*x^4)*Log[(1 + 5*
x^2)/x^2]^2)/(E^4*(676*x^2 + 3380*x^4) + E^4*(52*x^2 + 260*x^4)*Log[(1 + 5*x^2)/x^2] + E^4*(x^2 + 5*x^4)*Log[(
1 + 5*x^2)/x^2]^2),x]

[Out]

(x + 1/(x*(26 + Log[5 + x^(-2)])))/E^4

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fricas [B]  time = 0.88, size = 48, normalized size = 2.40 \begin {gather*} \frac {x^{2} \log \left (\frac {5 \, x^{2} + 1}{x^{2}}\right ) + 26 \, x^{2} + 1}{x e^{4} \log \left (\frac {5 \, x^{2} + 1}{x^{2}}\right ) + 26 \, x e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^4+x^2)*log((5*x^2+1)/x^2)^2+(260*x^4+47*x^2-1)*log((5*x^2+1)/x^2)+3380*x^4+546*x^2-24)/((5*x^4
+x^2)*exp(4)*log((5*x^2+1)/x^2)^2+(260*x^4+52*x^2)*exp(4)*log((5*x^2+1)/x^2)+(3380*x^4+676*x^2)*exp(4)),x, alg
orithm="fricas")

[Out]

(x^2*log((5*x^2 + 1)/x^2) + 26*x^2 + 1)/(x*e^4*log((5*x^2 + 1)/x^2) + 26*x*e^4)

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giac [B]  time = 0.37, size = 48, normalized size = 2.40 \begin {gather*} \frac {x^{2} \log \left (\frac {5 \, x^{2} + 1}{x^{2}}\right ) + 26 \, x^{2} + 1}{x e^{4} \log \left (\frac {5 \, x^{2} + 1}{x^{2}}\right ) + 26 \, x e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^4+x^2)*log((5*x^2+1)/x^2)^2+(260*x^4+47*x^2-1)*log((5*x^2+1)/x^2)+3380*x^4+546*x^2-24)/((5*x^4
+x^2)*exp(4)*log((5*x^2+1)/x^2)^2+(260*x^4+52*x^2)*exp(4)*log((5*x^2+1)/x^2)+(3380*x^4+676*x^2)*exp(4)),x, alg
orithm="giac")

[Out]

(x^2*log((5*x^2 + 1)/x^2) + 26*x^2 + 1)/(x*e^4*log((5*x^2 + 1)/x^2) + 26*x*e^4)

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maple [A]  time = 0.12, size = 28, normalized size = 1.40

method	result	size
risch	\(x \,{\mathrm e}^{-4}+\frac {{\mathrm e}^{-4}}{x \left (\ln \left (\frac {5 x^{2}+1}{x^{2}}\right )+26\right )}\)	\(28\)
norman	\(\frac {{\mathrm e}^{-4}+x^{2} {\mathrm e}^{-4} \ln \left (\frac {5 x^{2}+1}{x^{2}}\right )+26 x^{2} {\mathrm e}^{-4}}{x \left (\ln \left (\frac {5 x^{2}+1}{x^{2}}\right )+26\right )}\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((5*x^4+x^2)*ln((5*x^2+1)/x^2)^2+(260*x^4+47*x^2-1)*ln((5*x^2+1)/x^2)+3380*x^4+546*x^2-24)/((5*x^4+x^2)*ex
p(4)*ln((5*x^2+1)/x^2)^2+(260*x^4+52*x^2)*exp(4)*ln((5*x^2+1)/x^2)+(3380*x^4+676*x^2)*exp(4)),x,method=_RETURN
VERBOSE)

[Out]

x*exp(-4)+1/x*exp(-4)/(ln((5*x^2+1)/x^2)+26)

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maxima [B]  time = 0.45, size = 54, normalized size = 2.70 \begin {gather*} \frac {x^{2} \log \left (5 \, x^{2} + 1\right ) - 2 \, x^{2} \log \relax (x) + 26 \, x^{2} + 1}{x e^{4} \log \left (5 \, x^{2} + 1\right ) - 2 \, x e^{4} \log \relax (x) + 26 \, x e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^4+x^2)*log((5*x^2+1)/x^2)^2+(260*x^4+47*x^2-1)*log((5*x^2+1)/x^2)+3380*x^4+546*x^2-24)/((5*x^4
+x^2)*exp(4)*log((5*x^2+1)/x^2)^2+(260*x^4+52*x^2)*exp(4)*log((5*x^2+1)/x^2)+(3380*x^4+676*x^2)*exp(4)),x, alg
orithm="maxima")

[Out]

(x^2*log(5*x^2 + 1) - 2*x^2*log(x) + 26*x^2 + 1)/(x*e^4*log(5*x^2 + 1) - 2*x*e^4*log(x) + 26*x*e^4)

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mupad [B]  time = 4.48, size = 45, normalized size = 2.25 \begin {gather*} \frac {{\mathrm {e}}^{-4}\,\left (x^2\,\ln \left (\frac {5\,x^2+1}{x^2}\right )+26\,x^2+1\right )}{x\,\left (\ln \left (\frac {5\,x^2+1}{x^2}\right )+26\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log((5*x^2 + 1)/x^2)*(47*x^2 + 260*x^4 - 1) + log((5*x^2 + 1)/x^2)^2*(x^2 + 5*x^4) + 546*x^2 + 3380*x^4 -
24)/(exp(4)*(676*x^2 + 3380*x^4) + log((5*x^2 + 1)/x^2)*exp(4)*(52*x^2 + 260*x^4) + log((5*x^2 + 1)/x^2)^2*ex
p(4)*(x^2 + 5*x^4)),x)

[Out]

(exp(-4)*(x^2*log((5*x^2 + 1)/x^2) + 26*x^2 + 1))/(x*(log((5*x^2 + 1)/x^2) + 26))

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sympy [A]  time = 0.16, size = 29, normalized size = 1.45 \begin {gather*} \frac {x}{e^{4}} + \frac {1}{x e^{4} \log {\left (\frac {5 x^{2} + 1}{x^{2}} \right )} + 26 x e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x**4+x**2)*ln((5*x**2+1)/x**2)**2+(260*x**4+47*x**2-1)*ln((5*x**2+1)/x**2)+3380*x**4+546*x**2-24
)/((5*x**4+x**2)*exp(4)*ln((5*x**2+1)/x**2)**2+(260*x**4+52*x**2)*exp(4)*ln((5*x**2+1)/x**2)+(3380*x**4+676*x*
*2)*exp(4)),x)

[Out]

x*exp(-4) + 1/(x*exp(4)*log((5*x**2 + 1)/x**2) + 26*x*exp(4))

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